The probability for a central galaxy to interact with its
satellites depends on the number
of satellites within its host halo. Based on the
conditional stellar mass function (CSMF) for
satellites derived using group catalogs and
the abundance matching methods[1]}, [2]}, [3]},
we estimate the number of satellites with
stellar masses, \(M_{\rm s}\) ,
in the range of \([10^{-3}, 1]\times M_{\rm c}\) ,
where \(M_{\rm c}\)  is the stellar mass of the central galaxy,
and show the results in Figure REF .
The mean number of satellites per halo at \(\log {M_{\rm h}/\, h^{-1}{\rm M_\odot }}=12\)  is of
the order of unity at \(z\sim 0\) , and the variance among
different methods is not too large.
This means that central galaxies in these halos
have a high probability, close to unity,
to interact with their satellites with mass
above \(M_{\rm c}/1000\)  within an time scale, \(t_{\rm it}\) .
Since interaction requires that centrals and satellites
are close enough, \(t_{\rm it}\)  is expected to be comparable
to the merger timescale of galaxies. As shown in [4]},
merger time scale ranges from 1 Gyr to about 10 Gyrs,
with a typical value of 3 Gyrs.
